arXiv subjects

Refat Abdelmawla Khaled Assaad

Publications and source records attributed to Refat Abdelmawla Khaled Assaad.

4 recordsLinked to original sources

arXiv2023

On w-copure projective modules

Let $R$ be a commutative ring. An $R$-module $M$ is said to be $w$-split if Ext$_{R}^1(M,N)$ is a GV-torsion $R$-module for all $R$-modules $N$. It is known that every projective module is $w$-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called $w$-copure (resp., strongly $w$-copure) projective modules which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An $R$-module $M$ is said to be $w$-copure projective (resp., strongly $w$-copure projective) if Ext$_{R}^1(M,N)$ (resp., Ext$_{R}^n(M,N)$) is a GV-torsion $R$-module for all flat $R$-modules $N$ and any $n\geq1$.

math.AC↗

arXiv2023

On $τ_q$-weak global dimensions of commuative rings

In this paper, the $τ_q$-weak global dimension $τ_q$-\cwd$(R)$ of a commutative ring $R$ is introduced. Rings with $τ_q$-weak global dimension equal to $0$ are studied in terms of homologies, direct products, polynomial extensions and amalgamations. Besides, we investigate the $τ_q$-weak global dimensions of polynomial rings.

math.AC↗

arXiv2022

A note on weak w-projective modules

Let $R$ be a ring. An $R$-module $M$ is said to be a weak $w$-projective module if ${\rm Ext}_R^1(M,N)=0$ for all $N \in \mathcal{P}_{w}^{\dagger_\infty}$ (see, \cite{FLQ}). In this paper, we introduce and study some properties of weak $w$-projective modules. And we use these modules to characterize some classical rings, for example, we will prove that a ring $R$ is a $DW$-ring if and only if every weak $w$-projective is projective, $R$ is a Von Neumann regular ring if and only if every FP-projective is weak $w$-projective if and only if every finitely presented $R$-module is weak $w$-projective and $R$ is a $w$-semi-hereditary if and only if every finite type submodule of a free module is weak $w$-projective if and only if every finitely generated ideal of $R$ is a weak $w$-projective.

math.AC↗

arXiv2022

$w$-$\rm FP$-projective modules and dimension

Let $R$ be a ring. An $R$-module $M$ is said to be an absolutely $w$-pure module if and only if $\Ext^1_R(F,M)$ is a GV-torsion module for any finitely presented module $F$. In this paper, we introduce and study the concept of $w$-FP-projective module which is in some way a generalization of the notion of $\rm FP$-projective module. An $R$-module $M$ is said to be $w$-$\rm FP$-projective if $\Ext^1_R(M,N)=0$ for any absolutely $w$-pure module $N$. This new class of modules will be used to characterize (Noetherian) $DW$ rings. Hence, we introduce the $w$-$\rm FP$-projective dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. Illustrative examples are given.

math.AC↗